Wave Packet
A Gaussian wave packet propagates across the screen. Watch the group velocity carry the envelope while the phase velocity moves the carrier crests. Under dispersive media, the packet spreads — a direct manifestation of the uncertainty principle.
About this lab
A wave packet is a localized disturbance built from a superposition of sinusoidal waves with slightly different wavelengths. The envelope — the overall shape — travels at the group velocity vg = dω/dk, while the individual crests and troughs move at the phase velocity vp = ω/k. When these two velocities differ, you can see the carrier wave sliding through the envelope, crests appearing at one edge and vanishing at the other.
The crucial distinction between group and phase velocity was first clearly articulated by Lord Rayleigh in 1877, though the mathematics had been implicit in earlier work by Hamilton and others. Rayleigh noted that the velocity of energy transport in a dispersive medium is the group velocity, not the phase velocity. This insight was revolutionary: it meant that the speed of a signal — the speed at which information propagates — is in general different from the speed at which wave crests move.
In quantum mechanics, a free particle is described by a wave packet whose dispersion relation is ω = ℏk²/2m. Because ω depends quadratically on k, the group velocity vg = ℏk/m differs from the phase velocity vp = ℏk/2m by exactly a factor of two. More importantly, the quadratic dependence means the packet inevitably spreads over time: different Fourier components travel at different speeds, causing the envelope to broaden. This spreading is a direct physical manifestation of the Heisenberg uncertainty principle — a narrow packet in position space requires a broad spread in momentum space, and this momentum spread drives the spatial spreading.
There are remarkable exceptions to dispersive spreading. In certain nonlinear media, a wave packet can maintain its shape indefinitely, forming a soliton. First observed by John Scott Russell in 1834 as a solitary water wave propagating down a canal, solitons arise when nonlinear self-focusing exactly compensates for dispersive spreading. They appear in fiber optic communications, shallow water waves, plasma physics, and even in models of biological neural impulses. The soliton demonstrates that the tendency of wave packets to spread is not fundamental — it can be perfectly balanced by the right kind of nonlinearity.